Physics · No topic

To determine the value of x that makes vectors A and B perpendicular, we need to use the condition that the dot product of the two vectors must equal zero. The dot product of A = 2i + j + 3k and B = i + j + xk is calculated as follows:

A · B = (2)(1) + (1)(1) + (3)(x) = 2 + 1 + 3x = 3 + 3x.

Setting this equal to zero for perpendicularity gives:

3 + 3x = 0

3x = -3

x = -1.

Thus, the correct answer is -1. The other options do not satisfy the condition for perpendicularity, as substituting them results in a non-zero dot product.

Choosing x = 3 gives a dot product of 0, but it does not satisfy the perpendicularity condition when calculated.

Choosing x = -3 results in a non-zero dot product, indicating that the vectors are not perpendicular.

Setting x = 1 provides a non-zero result for the dot product, confirming the vectors are not perpendicular.

With x = -1, the dot product equals zero, confirming that vectors A and B are indeed perpendicular.